Problem 40 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(-x^{2}+2 x y+3 y^{2}, \quad x=-3\) and \(y=3\)

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Substitute Given Values

Start by substituting the given values of the variables into the expression. The given expression is \(-x^{2} + 2xy + 3y^{2}\). Substitute \(x = -3\) and \(y = 3\) into it.

Step 2: Compute Each Term

Calculate each part of the expression separately. - Compute \(-x^{2}\): \[ -(x^2) = -((-3)^2) = -9 \]- Compute \(2xy\): \[ 2xy = 2(-3)(3) = -18 \]- Compute \(3y^{2}\): \[ 3y^{2} = 3(3^2) = 27 \]

Step 3: Sum the Components

Now that each component of the expression is calculated, sum them together to find the final value.Combine the terms:\[-9 - 18 + 27 = 0 \]

Step 4: Final Answer

The final value of the expression when \(x = -3\) and \(y = 3\) is 0.

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in evaluating algebraic expressions. In doing so, we replace the variables in the expression with their given values. For example, in evaluating the expression \(-x^{2}+2xy+3y^{2}\), we substitute \(x = -3\) and \(y = 3\). This substitution turns our abstract expression into a numerical one, allowing us to perform straightforward arithmetic calculations.

Here's how you can apply the substitution method efficiently:

• Identify all the variables in the given algebraic expression.
• Look for the given values associated with these variables.
• Substitute the values into the expression wherever the corresponding variables appear.

Using the substitution method simplifies the expression and makes it ready for further evaluation. It's like putting numbers into a formula to see how everything adds up. Once substitution is completed, you proceed to calculate the results.

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial expression given specific variable values. A polynomial is an algebraic expression made up of terms including constants, variables, and exponents. For instance, \(-x^{2}+2xy+3y^{2}\) is a polynomial consisting of three terms.

In evaluating a polynomial:

• Substitute the given values of variables as shown in the substitution method.
• Simplify each term individually:
  • For \(-x^2\), calculate \(-((-3)^2)\), resulting in -9.
  • For \(2xy\), multiply \(2\), \(-3\), and \(3\) to get -18.
  • For \(3y^2\), compute \(3(3^2)\) which gives 27.
• Add up all the simplified terms to get the final result.

By evaluating each term clearly, you get a step-by-step understanding of how each part contributes to the whole expression. This process helps ensure that none of the mathematical procedures are skipped, leading to an accurate final answer.

Step-by-Step Calculation

Step-by-step calculation is a methodical approach to solving expressions that involves breaking down the process into manageable steps, ensuring accuracy at each phase. Let's see how this method works in evaluating the expression \(-x^{2}+2xy+3y^{2}\) with \(x = -3\) and \(y = 3\).

First, we start with substitution. Replace \(x\) and \(y\) with their values, transforming the expression into numerical terms. Then calculate each term individually:

• For \(-x^2\), compute \(-((-3)^2)\) which equals -9.
• For \(2xy\), compute \(2 \times (-3) \times 3\) yielding -18.
• For \(3y^2\), compute \(3 \times (3^2)\) which results in 27.

Now, add these computed values: \(-9 + (-18) + 27 = 0\). This summation leads directly to the answer.

Using step-by-step calculation:

• Makes it simpler to track each component of the expression.
• Helps to prevent mistakes by focusing on one operation at a time.
• Aids in understanding how changes in variables affect the outcome.

Practice makes perfect, and applying step-by-step calculation repeatedly will build confidence in handling increasingly complex algebraic expressions.